Shevchenko replaced Davide Ballardini as head coach of the Grifoni two weeks ago after a 2-2 draw at Empoli made it nine games without a win.
The Ukraine legend's first game in club management sees him hosting Roma and Mourinho, who was at the helm at Chelsea when 'Sheva' arrived at Stamford Bridge as a player in 2006.
The 45-year-old recently left his role in charge of Ukraine national team after a quarter-final defeat to England at UEFA Euro 2020.
Shevchenko enjoyed a very successful time as a player in Italy, scoring 127 goals in 226 Serie A games for AC Milan, and Mourinho expressed his hope that he can have success as a manager as well.
"He was an incredible player," Mourinho said on Sunday (AEDT) ahead of their meeting. "This is his first job in club management, but he did really well with Ukraine.
"I think he showed he has ideas, he has leadership... but I hope he loses the first of his club management career, like I did!"
Mourinho will also be looking to improve the fortunes of his own team, with Roma having won just one of its past seven games, and he will be missing several players through injury and COVID-19.
The Giallorossi announced that Bryan Cristante and Gonzalo Villar have tested positive for COVID-19, and join injured quartet Matias Vina, Chris Smalling, Riccardo Calafiori and Leonardo Spinazzola on the sidelines.
"We didn't do any training this morning. It will be fun for you to think about how we will play tomorrow," Mourinho said. "The work we've done in the week has to go in the bin a bit.
"It's clear we need to find new solutions and perhaps some players will have to 'sacrifice' themselves in different roles."
The former Real Madrid and Manchester United manager knows the risks that come with playing a team under a new head coach, and believes that factor will make the clash all the more interesting.
"At the very least, it's going to be an entertaining, intriguing game," he added. "The Genoa fans will be up for the game, the players will want to impress the new coach. It's a beautiful stadium and I'm sure it will be a great atmosphere."